April 1, 2011
Today’s problem is this:
Given an array X, find the j and i that maximizes Xj − Xi, subject to the condition that i ≤ j. If two different i,j pairs have equal differences, choose the “leftmost shortest” pair with the smallest i and, in case of a tie, the smallest j.
For instance, given an array [4, 3, 9, 1, 8, 2, 6, 7, 5], the maximum difference is 7 when i=3 and j=4. Given the array [4, 2, 9, 1, 8, 3, 6, 7, 5], the maximum difference of 7 appears at two points, but by the leftmost-shortest rule the desired result is i=1 and j=2. I and j need not be adjacent, as in the array [4, 3, 9, 1, 2, 6, 7, 8, 5], where the maximum difference of 7 is achieved when i=3 and j=7. If the array is monotonically decreasing the maximum difference is 0, which by the leftmost-shortest rule occurs when i=0 and j=0.
There are at least two solutions. The obvious solution that runs in quadratic time uses two nested loops, the outer loop over i from 0 to the length of the array n and the inner loop over j from i+1 to n, computing the difference between Xi and Xj and saving the result whenever a new maximum difference is found. There is also a clever linear-time solution that traverses the array once, simultaneously searching for a new minimum value and a new maximum difference; you’ll get it if you think about it for a minute.
Your task is to write both the quadratic and linear functions to compute the maximum difference in an array, and also a test function that demonstrates they are correct. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.